

<!DOCTYPE html>
<!--[if IE 8]><html class="no-js lt-ie9" lang="en" > <![endif]-->
<!--[if gt IE 8]><!--> <html class="no-js" lang="en" > <!--<![endif]-->
<head>
  <meta charset="utf-8">
  
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="Description" content="scikit-learn: machine learning in Python">

  
  <title>Single estimator versus bagging: bias-variance decomposition &mdash; scikit-learn 0.22 documentation</title>
  
  <link rel="canonical" href="http://scikit-learn.org/stable/auto_examples/ensemble/plot_bias_variance.html" />

  
  <link rel="shortcut icon" href="../../_static/favicon.ico"/>
  

  <link rel="stylesheet" href="../../_static/css/vendor/bootstrap.min.css" type="text/css" />
  <link rel="stylesheet" href="../../_static/gallery.css" type="text/css" />
  <link rel="stylesheet" href="../../_static/css/theme.css" type="text/css" />
<script id="documentation_options" data-url_root="../../" src="../../_static/documentation_options.js"></script>
<script src="../../_static/jquery.js"></script> 
</head>
<body>
<nav id="navbar" class="sk-docs-navbar navbar navbar-expand-md navbar-light bg-light py-0">
  <div class="container-fluid sk-docs-container px-0">
      <a class="navbar-brand py-0" href="../../index.html">
        <img
          class="sk-brand-img"
          src="../../_static/scikit-learn-logo-small.png"
          alt="logo"/>
      </a>
    <button
      id="sk-navbar-toggler"
      class="navbar-toggler"
      type="button"
      data-toggle="collapse"
      data-target="#navbarSupportedContent"
      aria-controls="navbarSupportedContent"
      aria-expanded="false"
      aria-label="Toggle navigation"
    >
      <span class="navbar-toggler-icon"></span>
    </button>

    <div class="sk-navbar-collapse collapse navbar-collapse" id="navbarSupportedContent">
      <ul class="navbar-nav mr-auto">
        <li class="nav-item">
          <a class="sk-nav-link nav-link" href="../../install.html">Install</a>
        </li>
        <li class="nav-item">
          <a class="sk-nav-link nav-link" href="../../user_guide.html">User Guide</a>
        </li>
        <li class="nav-item">
          <a class="sk-nav-link nav-link" href="../../modules/classes.html">API</a>
        </li>
        <li class="nav-item">
          <a class="sk-nav-link nav-link" href="../index.html">Examples</a>
        </li>
        <li class="nav-item">
          <a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../../getting_started.html">Getting Started</a>
        </li>
        <li class="nav-item">
          <a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../../tutorial/index.html">Tutorial</a>
        </li>
        <li class="nav-item">
          <a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../../glossary.html">Glossary</a>
        </li>
        <li class="nav-item">
          <a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../../developers/index.html">Development</a>
        </li>
        <li class="nav-item">
          <a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../../faq.html">FAQ</a>
        </li>
        <li class="nav-item">
          <a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../../related_projects.html">Related packages</a>
        </li>
        <li class="nav-item">
          <a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../../roadmap.html">Roadmap</a>
        </li>
        <li class="nav-item">
          <a class="sk-nav-link nav-link nav-more-item-mobile-items" href="../../about.html">About us</a>
        </li>
        <li class="nav-item">
          <a class="sk-nav-link nav-link nav-more-item-mobile-items" href="https://github.com/scikit-learn/scikit-learn">GitHub</a>
        </li>
        <li class="nav-item">
          <a class="sk-nav-link nav-link nav-more-item-mobile-items" href="https://scikit-learn.org/dev/versions.html">Other Versions</a>
        </li>
        <li class="nav-item dropdown nav-more-item-dropdown">
          <a class="sk-nav-link nav-link dropdown-toggle" href="#" id="navbarDropdown" role="button" data-toggle="dropdown" aria-haspopup="true" aria-expanded="false">More</a>
          <div class="dropdown-menu" aria-labelledby="navbarDropdown">
              <a class="sk-nav-dropdown-item dropdown-item" href="../../getting_started.html">Getting Started</a>
              <a class="sk-nav-dropdown-item dropdown-item" href="../../tutorial/index.html">Tutorial</a>
              <a class="sk-nav-dropdown-item dropdown-item" href="../../glossary.html">Glossary</a>
              <a class="sk-nav-dropdown-item dropdown-item" href="../../developers/index.html">Development</a>
              <a class="sk-nav-dropdown-item dropdown-item" href="../../faq.html">FAQ</a>
              <a class="sk-nav-dropdown-item dropdown-item" href="../../related_projects.html">Related packages</a>
              <a class="sk-nav-dropdown-item dropdown-item" href="../../roadmap.html">Roadmap</a>
              <a class="sk-nav-dropdown-item dropdown-item" href="../../about.html">About us</a>
              <a class="sk-nav-dropdown-item dropdown-item" href="https://github.com/scikit-learn/scikit-learn">GitHub</a>
              <a class="sk-nav-dropdown-item dropdown-item" href="https://scikit-learn.org/dev/versions.html">Other Versions</a>
          </div>
        </li>
      </ul>
      <div id="searchbox" role="search">
          <div class="searchformwrapper">
          <form class="search" action="../../search.html" method="get">
            <input class="sk-search-text-input" type="text" name="q" aria-labelledby="searchlabel" />
            <input class="sk-search-text-btn" type="submit" value="Go" />
          </form>
          </div>
      </div>
    </div>
  </div>
</nav>
<div class="d-flex" id="sk-doc-wrapper">
    <input type="checkbox" name="sk-toggle-checkbox" id="sk-toggle-checkbox">
    <label id="sk-sidemenu-toggle" class="sk-btn-toggle-toc btn sk-btn-primary" for="sk-toggle-checkbox">Toggle Menu</label>
    <div id="sk-sidebar-wrapper" class="border-right">
      <div class="sk-sidebar-toc-wrapper">
        <div class="sk-sidebar-toc-logo">
          <a href="../../index.html">
            <img
              class="sk-brand-img"
              src="../../_static/scikit-learn-logo-small.png"
              alt="logo"/>
          </a>
        </div>
        <div class="btn-group w-100 mb-2" role="group" aria-label="rellinks">
            <a href="plot_gradient_boosting_oob.html" role="button" class="btn sk-btn-rellink py-1" sk-rellink-tooltip="Gradient Boosting Out-of-Bag estimates">Prev</a><a href="../index.html" role="button" class="btn sk-btn-rellink py-1" sk-rellink-tooltip="Examples">Up</a>
            <a href="plot_forest_iris.html" role="button" class="btn sk-btn-rellink py-1" sk-rellink-tooltip="Plot the decision surfaces of ensembles of trees on the iris dataset">Next</a>
        </div>
        <div class="alert alert-danger p-1 mb-2" role="alert">
          <p class="text-center mb-0">
          <strong>scikit-learn 0.22</strong><br/>
          <a href="http://scikit-learn.org/dev/versions.html">Other versions</a>
          </p>
        </div>
        <div class="alert alert-warning p-1 mb-2" role="alert">
          <p class="text-center mb-0">
            Please <a class="font-weight-bold" href="../../about.html#citing-scikit-learn"><string>cite us</string></a> if you use the software.
          </p>
        </div>
          <div class="sk-sidebar-toc">
            <ul>
<li><a class="reference internal" href="#">Single estimator versus bagging: bias-variance decomposition</a><ul>
<li><a class="reference internal" href="#references">References</a></li>
</ul>
</li>
</ul>

          </div>
      </div>
    </div>
    <div id="sk-page-content-wrapper">
      <div class="sk-page-content container-fluid body px-md-3" role="main">
        
  <div class="sphx-glr-download-link-note admonition note">
<p class="admonition-title">Note</p>
<p>Click <a class="reference internal" href="#sphx-glr-download-auto-examples-ensemble-plot-bias-variance-py"><span class="std std-ref">here</span></a> to download the full example code or to run this example in your browser via Binder</p>
</div>
<div class="sphx-glr-example-title section" id="single-estimator-versus-bagging-bias-variance-decomposition">
<span id="sphx-glr-auto-examples-ensemble-plot-bias-variance-py"></span><h1>Single estimator versus bagging: bias-variance decomposition<a class="headerlink" href="#single-estimator-versus-bagging-bias-variance-decomposition" title="Permalink to this headline">¶</a></h1>
<p>This example illustrates and compares the bias-variance decomposition of the
expected mean squared error of a single estimator against a bagging ensemble.</p>
<p>In regression, the expected mean squared error of an estimator can be
decomposed in terms of bias, variance and noise. On average over datasets of
the regression problem, the bias term measures the average amount by which the
predictions of the estimator differ from the predictions of the best possible
estimator for the problem (i.e., the Bayes model). The variance term measures
the variability of the predictions of the estimator when fit over different
instances LS of the problem. Finally, the noise measures the irreducible part
of the error which is due the variability in the data.</p>
<p>The upper left figure illustrates the predictions (in dark red) of a single
decision tree trained over a random dataset LS (the blue dots) of a toy 1d
regression problem. It also illustrates the predictions (in light red) of other
single decision trees trained over other (and different) randomly drawn
instances LS of the problem. Intuitively, the variance term here corresponds to
the width of the beam of predictions (in light red) of the individual
estimators. The larger the variance, the more sensitive are the predictions for
<code class="docutils literal notranslate"><span class="pre">x</span></code> to small changes in the training set. The bias term corresponds to the
difference between the average prediction of the estimator (in cyan) and the
best possible model (in dark blue). On this problem, we can thus observe that
the bias is quite low (both the cyan and the blue curves are close to each
other) while the variance is large (the red beam is rather wide).</p>
<p>The lower left figure plots the pointwise decomposition of the expected mean
squared error of a single decision tree. It confirms that the bias term (in
blue) is low while the variance is large (in green). It also illustrates the
noise part of the error which, as expected, appears to be constant and around
<code class="docutils literal notranslate"><span class="pre">0.01</span></code>.</p>
<p>The right figures correspond to the same plots but using instead a bagging
ensemble of decision trees. In both figures, we can observe that the bias term
is larger than in the previous case. In the upper right figure, the difference
between the average prediction (in cyan) and the best possible model is larger
(e.g., notice the offset around <code class="docutils literal notranslate"><span class="pre">x=2</span></code>). In the lower right figure, the bias
curve is also slightly higher than in the lower left figure. In terms of
variance however, the beam of predictions is narrower, which suggests that the
variance is lower. Indeed, as the lower right figure confirms, the variance
term (in green) is lower than for single decision trees. Overall, the bias-
variance decomposition is therefore no longer the same. The tradeoff is better
for bagging: averaging several decision trees fit on bootstrap copies of the
dataset slightly increases the bias term but allows for a larger reduction of
the variance, which results in a lower overall mean squared error (compare the
red curves int the lower figures). The script output also confirms this
intuition. The total error of the bagging ensemble is lower than the total
error of a single decision tree, and this difference indeed mainly stems from a
reduced variance.</p>
<p>For further details on bias-variance decomposition, see section 7.3 of <a class="footnote-reference brackets" href="#id2" id="id1">1</a>.</p>
<div class="section" id="references">
<h2>References<a class="headerlink" href="#references" title="Permalink to this headline">¶</a></h2>
<dl class="footnote brackets">
<dt class="label" id="id2"><span class="brackets"><a class="fn-backref" href="#id1">1</a></span></dt>
<dd><p>T. Hastie, R. Tibshirani and J. Friedman,
“Elements of Statistical Learning”, Springer, 2009.</p>
</dd>
</dl>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="vm">__doc__</span><span class="p">)</span>

<span class="c1"># Author: Gilles Louppe &lt;g.louppe@gmail.com&gt;</span>
<span class="c1"># License: BSD 3 clause</span>

<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>

<span class="kn">from</span> <span class="nn">sklearn.ensemble</span> <span class="kn">import</span> <span class="n">BaggingRegressor</span>
<span class="kn">from</span> <span class="nn">sklearn.tree</span> <span class="kn">import</span> <span class="n">DecisionTreeRegressor</span>

<span class="c1"># Settings</span>
<span class="n">n_repeat</span> <span class="o">=</span> <span class="mi">50</span>       <span class="c1"># Number of iterations for computing expectations</span>
<span class="n">n_train</span> <span class="o">=</span> <span class="mi">50</span>        <span class="c1"># Size of the training set</span>
<span class="n">n_test</span> <span class="o">=</span> <span class="mi">1000</span>       <span class="c1"># Size of the test set</span>
<span class="n">noise</span> <span class="o">=</span> <span class="mf">0.1</span>         <span class="c1"># Standard deviation of the noise</span>
<span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span>

<span class="c1"># Change this for exploring the bias-variance decomposition of other</span>
<span class="c1"># estimators. This should work well for estimators with high variance (e.g.,</span>
<span class="c1"># decision trees or KNN), but poorly for estimators with low variance (e.g.,</span>
<span class="c1"># linear models).</span>
<span class="n">estimators</span> <span class="o">=</span> <span class="p">[(</span><span class="s2">&quot;Tree&quot;</span><span class="p">,</span> <span class="n">DecisionTreeRegressor</span><span class="p">()),</span>
              <span class="p">(</span><span class="s2">&quot;Bagging(Tree)&quot;</span><span class="p">,</span> <span class="n">BaggingRegressor</span><span class="p">(</span><span class="n">DecisionTreeRegressor</span><span class="p">()))]</span>

<span class="n">n_estimators</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="n">estimators</span><span class="p">)</span>


<span class="c1"># Generate data</span>
<span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
    <span class="n">x</span> <span class="o">=</span> <span class="n">x</span><span class="o">.</span><span class="n">ravel</span><span class="p">()</span>

    <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span> <span class="o">+</span> <span class="mf">1.5</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="p">(</span><span class="n">x</span> <span class="o">-</span> <span class="mi">2</span><span class="p">)</span> <span class="o">**</span> <span class="mi">2</span><span class="p">)</span>


<span class="k">def</span> <span class="nf">generate</span><span class="p">(</span><span class="n">n_samples</span><span class="p">,</span> <span class="n">noise</span><span class="p">,</span> <span class="n">n_repeat</span><span class="o">=</span><span class="mi">1</span><span class="p">):</span>
    <span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="n">n_samples</span><span class="p">)</span> <span class="o">*</span> <span class="mi">10</span> <span class="o">-</span> <span class="mi">5</span>
    <span class="n">X</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">sort</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>

    <span class="k">if</span> <span class="n">n_repeat</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
        <span class="n">y</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">X</span><span class="p">)</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="mf">0.0</span><span class="p">,</span> <span class="n">noise</span><span class="p">,</span> <span class="n">n_samples</span><span class="p">)</span>
    <span class="k">else</span><span class="p">:</span>
        <span class="n">y</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n_samples</span><span class="p">,</span> <span class="n">n_repeat</span><span class="p">))</span>

        <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_repeat</span><span class="p">):</span>
            <span class="n">y</span><span class="p">[:,</span> <span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">X</span><span class="p">)</span> <span class="o">+</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">normal</span><span class="p">(</span><span class="mf">0.0</span><span class="p">,</span> <span class="n">noise</span><span class="p">,</span> <span class="n">n_samples</span><span class="p">)</span>

    <span class="n">X</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">reshape</span><span class="p">((</span><span class="n">n_samples</span><span class="p">,</span> <span class="mi">1</span><span class="p">))</span>

    <span class="k">return</span> <span class="n">X</span><span class="p">,</span> <span class="n">y</span>


<span class="n">X_train</span> <span class="o">=</span> <span class="p">[]</span>
<span class="n">y_train</span> <span class="o">=</span> <span class="p">[]</span>

<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_repeat</span><span class="p">):</span>
    <span class="n">X</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">generate</span><span class="p">(</span><span class="n">n_samples</span><span class="o">=</span><span class="n">n_train</span><span class="p">,</span> <span class="n">noise</span><span class="o">=</span><span class="n">noise</span><span class="p">)</span>
    <span class="n">X_train</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
    <span class="n">y_train</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">y</span><span class="p">)</span>

<span class="n">X_test</span><span class="p">,</span> <span class="n">y_test</span> <span class="o">=</span> <span class="n">generate</span><span class="p">(</span><span class="n">n_samples</span><span class="o">=</span><span class="n">n_test</span><span class="p">,</span> <span class="n">noise</span><span class="o">=</span><span class="n">noise</span><span class="p">,</span> <span class="n">n_repeat</span><span class="o">=</span><span class="n">n_repeat</span><span class="p">)</span>

<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">(</span><span class="n">figsize</span><span class="o">=</span><span class="p">(</span><span class="mi">10</span><span class="p">,</span> <span class="mi">8</span><span class="p">))</span>

<span class="c1"># Loop over estimators to compare</span>
<span class="k">for</span> <span class="n">n</span><span class="p">,</span> <span class="p">(</span><span class="n">name</span><span class="p">,</span> <span class="n">estimator</span><span class="p">)</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">estimators</span><span class="p">):</span>
    <span class="c1"># Compute predictions</span>
    <span class="n">y_predict</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="n">n_test</span><span class="p">,</span> <span class="n">n_repeat</span><span class="p">))</span>

    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_repeat</span><span class="p">):</span>
        <span class="n">estimator</span><span class="o">.</span><span class="n">fit</span><span class="p">(</span><span class="n">X_train</span><span class="p">[</span><span class="n">i</span><span class="p">],</span> <span class="n">y_train</span><span class="p">[</span><span class="n">i</span><span class="p">])</span>
        <span class="n">y_predict</span><span class="p">[:,</span> <span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">estimator</span><span class="o">.</span><span class="n">predict</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span>

    <span class="c1"># Bias^2 + Variance + Noise decomposition of the mean squared error</span>
    <span class="n">y_error</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">(</span><span class="n">n_test</span><span class="p">)</span>

    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_repeat</span><span class="p">):</span>
        <span class="k">for</span> <span class="n">j</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_repeat</span><span class="p">):</span>
            <span class="n">y_error</span> <span class="o">+=</span> <span class="p">(</span><span class="n">y_test</span><span class="p">[:,</span> <span class="n">j</span><span class="p">]</span> <span class="o">-</span> <span class="n">y_predict</span><span class="p">[:,</span> <span class="n">i</span><span class="p">])</span> <span class="o">**</span> <span class="mi">2</span>

    <span class="n">y_error</span> <span class="o">/=</span> <span class="p">(</span><span class="n">n_repeat</span> <span class="o">*</span> <span class="n">n_repeat</span><span class="p">)</span>

    <span class="n">y_noise</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">var</span><span class="p">(</span><span class="n">y_test</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
    <span class="n">y_bias</span> <span class="o">=</span> <span class="p">(</span><span class="n">f</span><span class="p">(</span><span class="n">X_test</span><span class="p">)</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">y_predict</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">))</span> <span class="o">**</span> <span class="mi">2</span>
    <span class="n">y_var</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">var</span><span class="p">(</span><span class="n">y_predict</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>

    <span class="nb">print</span><span class="p">(</span><span class="s2">&quot;</span><span class="si">{0}</span><span class="s2">: </span><span class="si">{1:.4f}</span><span class="s2"> (error) = </span><span class="si">{2:.4f}</span><span class="s2"> (bias^2) &quot;</span>
          <span class="s2">&quot; + </span><span class="si">{3:.4f}</span><span class="s2"> (var) + </span><span class="si">{4:.4f}</span><span class="s2"> (noise)&quot;</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">name</span><span class="p">,</span>
                                                      <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">y_error</span><span class="p">),</span>
                                                      <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">y_bias</span><span class="p">),</span>
                                                      <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">y_var</span><span class="p">),</span>
                                                      <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">y_noise</span><span class="p">)))</span>

    <span class="c1"># Plot figures</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">n_estimators</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span> <span class="n">f</span><span class="p">(</span><span class="n">X_test</span><span class="p">),</span> <span class="s2">&quot;b&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;$f(x)$&quot;</span><span class="p">)</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X_train</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">y_train</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="s2">&quot;.b&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;LS ~ $y = f(x)+noise$&quot;</span><span class="p">)</span>

    <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_repeat</span><span class="p">):</span>
        <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
            <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span> <span class="n">y_predict</span><span class="p">[:,</span> <span class="n">i</span><span class="p">],</span> <span class="s2">&quot;r&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="sa">r</span><span class="s2">&quot;$\^y(x)$&quot;</span><span class="p">)</span>
        <span class="k">else</span><span class="p">:</span>
            <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span> <span class="n">y_predict</span><span class="p">[:,</span> <span class="n">i</span><span class="p">],</span> <span class="s2">&quot;r&quot;</span><span class="p">,</span> <span class="n">alpha</span><span class="o">=</span><span class="mf">0.05</span><span class="p">)</span>

    <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">y_predict</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">),</span> <span class="s2">&quot;c&quot;</span><span class="p">,</span>
             <span class="n">label</span><span class="o">=</span><span class="sa">r</span><span class="s2">&quot;$\mathbb</span><span class="si">{E}</span><span class="s2">_</span><span class="si">{LS}</span><span class="s2"> \^y(x)$&quot;</span><span class="p">)</span>

    <span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">([</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="n">name</span><span class="p">)</span>

    <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="n">n_estimators</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>
        <span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="p">(</span><span class="mf">1.1</span><span class="p">,</span> <span class="o">.</span><span class="mi">5</span><span class="p">))</span>

    <span class="n">plt</span><span class="o">.</span><span class="n">subplot</span><span class="p">(</span><span class="mi">2</span><span class="p">,</span> <span class="n">n_estimators</span><span class="p">,</span> <span class="n">n_estimators</span> <span class="o">+</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span> <span class="n">y_error</span><span class="p">,</span> <span class="s2">&quot;r&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;$error(x)$&quot;</span><span class="p">)</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span> <span class="n">y_bias</span><span class="p">,</span> <span class="s2">&quot;b&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;$bias^2(x)$&quot;</span><span class="p">),</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span> <span class="n">y_var</span><span class="p">,</span> <span class="s2">&quot;g&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;$variance(x)$&quot;</span><span class="p">),</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">X_test</span><span class="p">,</span> <span class="n">y_noise</span><span class="p">,</span> <span class="s2">&quot;c&quot;</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">&quot;$noise(x)$&quot;</span><span class="p">)</span>

    <span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">([</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">])</span>
    <span class="n">plt</span><span class="o">.</span><span class="n">ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">])</span>

    <span class="k">if</span> <span class="n">n</span> <span class="o">==</span> <span class="n">n_estimators</span> <span class="o">-</span> <span class="mi">1</span><span class="p">:</span>

        <span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="p">(</span><span class="mf">1.1</span><span class="p">,</span> <span class="o">.</span><span class="mi">5</span><span class="p">))</span>

<span class="n">plt</span><span class="o">.</span><span class="n">subplots_adjust</span><span class="p">(</span><span class="n">right</span><span class="o">=.</span><span class="mi">75</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
</pre></div>
</div>
<p class="sphx-glr-timing"><strong>Total running time of the script:</strong> ( 0 minutes  0.000 seconds)</p>
<div class="sphx-glr-footer class sphx-glr-footer-example docutils container" id="sphx-glr-download-auto-examples-ensemble-plot-bias-variance-py">
<div class="binder-badge docutils container">
<a class="reference external image-reference" href="https://mybinder.org/v2/gh/scikit-learn/scikit-learn/0.22.X?urlpath=lab/tree/notebooks/auto_examples/ensemble/plot_bias_variance.ipynb"><img alt="https://mybinder.org/badge_logo.svg" src="https://mybinder.org/badge_logo.svg" width="150px" /></a>
</div>
<div class="sphx-glr-download docutils container">
<p><a class="reference download internal" download="" href="../../_downloads/c195b57124cc243ae523d3445f602535/plot_bias_variance.py"><code class="xref download docutils literal notranslate"><span class="pre">Download</span> <span class="pre">Python</span> <span class="pre">source</span> <span class="pre">code:</span> <span class="pre">plot_bias_variance.py</span></code></a></p>
</div>
<div class="sphx-glr-download docutils container">
<p><a class="reference download internal" download="" href="../../_downloads/8eea81d03a067e96d18eca0abd38da53/plot_bias_variance.ipynb"><code class="xref download docutils literal notranslate"><span class="pre">Download</span> <span class="pre">Jupyter</span> <span class="pre">notebook:</span> <span class="pre">plot_bias_variance.ipynb</span></code></a></p>
</div>
</div>
<p class="sphx-glr-signature"><a class="reference external" href="https://sphinx-gallery.github.io">Gallery generated by Sphinx-Gallery</a></p>
</div>
</div>


      </div>
    <div class="container">
      <footer class="sk-content-footer">
            &copy; 2007 - 2019, scikit-learn developers (BSD License).
          <a href="../../_sources/auto_examples/ensemble/plot_bias_variance.rst.txt" rel="nofollow">Show this page source</a>
      </footer>
    </div>
  </div>
</div>
<script src="../../_static/js/vendor/bootstrap.min.js"></script>

<script>
    window.ga=window.ga||function(){(ga.q=ga.q||[]).push(arguments)};ga.l=+new Date;
    ga('create', 'UA-22606712-2', 'auto');
    ga('set', 'anonymizeIp', true);
    ga('send', 'pageview');
</script>
<script async src='https://www.google-analytics.com/analytics.js'></script>


<script>
$(document).ready(function() {
    /* Add a [>>>] button on the top-right corner of code samples to hide
     * the >>> and ... prompts and the output and thus make the code
     * copyable. */
    var div = $('.highlight-python .highlight,' +
                '.highlight-python3 .highlight,' +
                '.highlight-pycon .highlight,' +
		'.highlight-default .highlight')
    var pre = div.find('pre');

    // get the styles from the current theme
    pre.parent().parent().css('position', 'relative');
    var hide_text = 'Hide prompts and outputs';
    var show_text = 'Show prompts and outputs';

    // create and add the button to all the code blocks that contain >>>
    div.each(function(index) {
        var jthis = $(this);
        if (jthis.find('.gp').length > 0) {
            var button = $('<span class="copybutton">&gt;&gt;&gt;</span>');
            button.attr('title', hide_text);
            button.data('hidden', 'false');
            jthis.prepend(button);
        }
        // tracebacks (.gt) contain bare text elements that need to be
        // wrapped in a span to work with .nextUntil() (see later)
        jthis.find('pre:has(.gt)').contents().filter(function() {
            return ((this.nodeType == 3) && (this.data.trim().length > 0));
        }).wrap('<span>');
    });

    // define the behavior of the button when it's clicked
    $('.copybutton').click(function(e){
        e.preventDefault();
        var button = $(this);
        if (button.data('hidden') === 'false') {
            // hide the code output
            button.parent().find('.go, .gp, .gt').hide();
            button.next('pre').find('.gt').nextUntil('.gp, .go').css('visibility', 'hidden');
            button.css('text-decoration', 'line-through');
            button.attr('title', show_text);
            button.data('hidden', 'true');
        } else {
            // show the code output
            button.parent().find('.go, .gp, .gt').show();
            button.next('pre').find('.gt').nextUntil('.gp, .go').css('visibility', 'visible');
            button.css('text-decoration', 'none');
            button.attr('title', hide_text);
            button.data('hidden', 'false');
        }
    });

	/*** Add permalink buttons next to glossary terms ***/
	$('dl.glossary > dt[id]').append(function() {
		return ('<a class="headerlink" href="#' +
			    this.getAttribute('id') +
			    '" title="Permalink to this term">¶</a>');
	});
  /*** Hide navbar when scrolling down ***/
  // Returns true when headerlink target matches hash in url
  (function() {
    hashTargetOnTop = function() {
        var hash = window.location.hash;
        if ( hash.length < 2 ) { return false; }

        var target = document.getElementById( hash.slice(1) );
        if ( target === null ) { return false; }

        var top = target.getBoundingClientRect().top;
        return (top < 2) && (top > -2);
    };

    // Hide navbar on load if hash target is on top
    var navBar = document.getElementById("navbar");
    var navBarToggler = document.getElementById("sk-navbar-toggler");
    var navBarHeightHidden = "-" + navBar.getBoundingClientRect().height + "px";
    var $window = $(window);

    hideNavBar = function() {
        navBar.style.top = navBarHeightHidden;
    };

    showNavBar = function() {
        navBar.style.top = "0";
    }

    if (hashTargetOnTop()) {
        hideNavBar()
    }

    var prevScrollpos = window.pageYOffset;
    hideOnScroll = function(lastScrollTop) {
        if (($window.width() < 768) && (navBarToggler.getAttribute("aria-expanded") === 'true')) {
            return;
        }
        if (lastScrollTop > 2 && (prevScrollpos <= lastScrollTop) || hashTargetOnTop()){
            hideNavBar()
        } else {
            showNavBar()
        }
        prevScrollpos = lastScrollTop;
    };

    /*** high preformance scroll event listener***/
    var raf = window.requestAnimationFrame ||
        window.webkitRequestAnimationFrame ||
        window.mozRequestAnimationFrame ||
        window.msRequestAnimationFrame ||
        window.oRequestAnimationFrame;
    var lastScrollTop = $window.scrollTop();

    if (raf) {
        loop();
    }

    function loop() {
        var scrollTop = $window.scrollTop();
        if (lastScrollTop === scrollTop) {
            raf(loop);
            return;
        } else {
            lastScrollTop = scrollTop;
            hideOnScroll(lastScrollTop);
            raf(loop);
        }
    }
  })();
});

</script>
    
<script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js"></script>
    
</body>
</html>